The Gamma-Switch Ramsey Number

Abstract

We define and develop preliminary theoretical results for the $\hat{I}“$-switch Ramsey number, a variation on the classical $m$-colour Ramsey number for which we allow permuting the colours incident with a vertex using elements of a group $\hat{I}“\le S_m$. We find bounds for the $\hat{I}“$-switch Ramsey number for groups with various properties as a function of the classical parameter. We prove $R_{C_3}(4,4,4)=R(3,3,3)+1$, $R_{C_4}(3,4,3,4)=R(2,3,2,3)+1$ and $43\le R_{C_4}(4,4,4,4)\le R(3,3,3,3)+1$.

15 pages